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\title{Boy's Surface (After Ap\'{e}ry)}
\author{Hermann Karcher}
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The Boy's Surface is really a family of surfaces. Werner Boy, in his dissertation
under Hilbert, constructed an immersion of the projective plane. (Being
nonorientable implies that no embedding is possible.) Boy's Surface has
besides its self-intersection curves only one more serious singularity,
namely a triple point. His construction was topological.

Ap\'{e}ry found algebraically embedded ``Boy's Surfaces''. They carry
one-parameter families of ellipses.

The Bryant-Kusner Boy's Surfaces are obtained by an inversion from
a minimal surface in $R^3$.  This minimal surface is an immersion
of $S^2 - 6$ points such that antipodal points have the same image 
in $R^3$.  The six punctures are three antipodal pairs, and the 
minimal surface has so called "planar ends" at these punctures.
In this context it is important that the puncture in the inversion 
of a planar end can be smoothly closed by adding a point. The 
closing of the three pairs of antipodal ends thus gives a triple
point on the surface obtained by inversion.

The formulae for Ap\'{e}ry's immersion are as follows:

$\mathrm{x}(u,v) = (2/3) \frac{(\cos(u) \cos(2 v)+\sqrt 2 \sin(u) \cos(v)) \cos(u)}{\sqrt 2 - \sin(2 u) \sin(3 v)}$

$\mathrm{y}(u,v) = (2/3) \frac{(\cos(u) \sin(2 v)-\sqrt 2 \sin(u) \sin(v)) \cos(u)}{\sqrt 2-\sin(2 u) \sin(3 v)}$

$\mathrm{z}(u,v) = \frac{\sqrt 2 \cos(u)^2} { (\sqrt 2 - \sin(2 u) \sin(2 v))}$

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